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- 저자 : 장규환 (검색결과 126 건)
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Integral domains with a free semigroup of *-invertible integral *-ideals
장규환,김환구 대한수학회 2011 대한수학회보 Vol.48 No.6
Let * be a star-operation on an integral domain R, and let [기호](R) be the semigroup of *-invertible integral *-ideals of R. In this article, we introduce the concept of a *-coatom, and we then characterize when [기호](R) is a free semigroup with a set of free generators consisting of *-coatoms. In particular, we show that [기호](R) is a free semigroup if and only if R is a Krull domain and each v-invertible v-ideal is *-invertible. As a corollary, we obtain some characterizations of *-Dedekind domains. Let * be a star-operation on an integral domain R, and let [기호](R) be the semigroup of *-invertible integral *-ideals of R. In this article, we introduce the concept of a *-coatom, and we then characterize when [기호](R) is a free semigroup with a set of free generators consisting of *-coatoms. In particular, we show that [기호](R) is a free semigroup if and only if R is a Krull domain and each v-invertible v-ideal is *-invertible. As a corollary, we obtain some characterizations of *-Dedekind domains.
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Strong Mori modules over an integral domain
장규환 대한수학회 2013 대한수학회보 Vol.50 No.6
Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and Nv = {f∈D[X] | c(f)v = D}. Let q(M) = M D K and MwD = {x ∈ q(M) | xJ ⊆ M for a nonzero finitely generated ideal J of D with Jv = D}. In this paper, we show that MwD = M[X]Nv ∩ q(M) and (M[X])w D[X] ∩ q(M)[X] = MwD[X] = M[X]Nv ∩ q(M)[X]. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]- module if and only if M[X]Nv is a Noetherian D[X]Nv -module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if D[X]Nv is a Noetherian domain.
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Two generalizations of LCM-stable extensions
장규환,김환구,임정욱 대한수학회 2013 대한수학회지 Vol.50 No.2
Let R ⊆ T be an extension of integral domains, X be anindeterminate over T, and R[X] and T[X] be polynomial rings. Then R ⊆ T is said to be LCM-stable if (aR∩bR)T = aT∩bT for all 0 ≠ a, b ∈ R. Let wA be the so-called w-operation on an integral domain A. In this paper,we introduce the notions of w(e)- and w-LCMstable extensions: (i) R ⊆ Tis w(e)-LCM-stable if ((aR ∩ bR)T)wT = aT ∩ bT for all 0 ≠ a, b ∈ R and (ii) R ⊆ T is w-LCM-stable if ((aR ∩ bR)T)wR = (aT ∩ bT)wR for all 0 ≠ a, b 2 R. We prove that LCM-stable extensions are both w(e)-LCM-stable and w-LCM-stable. We also generalize some results on LCM-stableextensions. Among other things, we show that if R is a Krull domain(resp., PvMD), then R ⊆ T is w(e)-LCM-stable (resp., w-LCM-stable) ifand only if R[X] ⊆ T[X] is w(e)-LCM-stable (resp., w-LCM-stable).
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UPPERS TO ZERO IN POLYNOMIAL RINGS WHICH ARE MAXIMAL IDEALS
장규환 대한수학회 2015 대한수학회보 Vol.52 No.2
Let D be an integrally closed domain with quotient field K, X be an indeterminate over D, f = a0 + a1X + · · · + anXn ∈ D[X] be irreducible in K[X], and Qf = fK[X] ∩ D[X]. In this paper, we show that Qf is a maximal ideal of D[X] if and only if ( a1/a0 , . . . , an/a0 ) ⊆ P for all nonzero prime ideals P of D; in this case, Qf = 1/a0 fD[X]. As a corollary, we have that if D is a Krull domain, then D has infinitely many height- one prime ideals if and only if each maximal ideal of D[X] has height ≥ 2.
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KAPLANSKY-TYPE THEOREMS IN GRADED INTEGRAL DOMAINS
장규환,김환구,오동렬 대한수학회 2015 대한수학회보 Vol.52 No.4
It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky’s theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, B´ezout domain, valuation domain, Krull domain, π-domain).
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THE KRONECKER FUNCTION RING OF THE RING D[X]N
장규환 대한수학회 2010 대한수학회보 Vol.47 No.5
Let D be an integrally closed domain with quotient field K,* be a star operation on D, X, Y be indeterminates over D, N* = {f ∈D[X]| (cD(f))* = D} and R = D[X]N* . Let b be the b-operation on R,and let *c be the star operation on D defined by I*c = (ID[X]N* )b ∩K. Finally, let Kr(R, b) (resp., Kr(D, *c)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) ⊆ Kr(D, *c) and Kr(R, b) is a kfr with respect to K(Y ) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, *c) if and only if D is a P*MD. As a corollary, we have that if D is not a P*MD, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y ) and X.
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LOCALLY PSEUDO-VALUATION DOMAINS OF THE FORM D[X]Nv
장규환 대한수학회 2008 대한수학회지 Vol.45 No.5
Let D be an integral domain, X an indeterminate over D, Nv = {f ∈ D[X]|(Af )v = D}. Among other things, we introduce the concept of t-locally PVDs and prove that D[X]Nv is a locally PVD if and only if D is a t-locally PVD and a UMT-domain, if and only if D[X] is a t-locally PVD, if and only if each overring of D[X]Nv is a locally PVD. Let D be an integral domain, X an indeterminate over D, Nv = {f ∈ D[X]|(Af )v = D}. Among other things, we introduce the concept of t-locally PVDs and prove that D[X]Nv is a locally PVD if and only if D is a t-locally PVD and a UMT-domain, if and only if D[X] is a t-locally PVD, if and only if each overring of D[X]Nv is a locally PVD.
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Graded integral domains in which each nonzero homogeneous ideal is divisorial
장규환,Haleh Hamdi,Parviz Sahandi 대한수학회 2019 대한수학회보 Vol.56 No.4
Let $\Gamma$ be a nonzero commutative cancellative monoid (written additively), $R = \bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain with $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, and $S(H) = \{f \in R \,|\, C(f) = R\}$. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if $R$ is integrally closed, then $R$ is a homogeneously divisorial domain if and only if $R_{S(H)}$ is an h-local Pr\"ufer domain whose maximal ideals are invertible, if and only if $R$ satisfies the following four conditions: (i) $R$ is a graded-Pr\"{u}fer domain, (ii) every homogeneous maximal ideal of $R$ is invertible, (iii) each nonzero homogeneous prime ideal of $R$ is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of $R$ has only finitely many minimal prime ideals. We also show that if $R$ is a graded-Noetherian domain, then $R$ is a homogeneously divisorial domain if and only if $R_{S(H)}$ is a divisorial domain of (Krull) dimension one.
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